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Shared Qs (029)


  1. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-6.3,0)\) and \((6.3,0)\) and a covertex at \((0, 6)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (4.7, 5.05):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  2. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-4,0)\) and \((4,0)\) and a covertex at \((0, 4.2)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (4.21, 2.89):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  3. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-3.6,0)\) and \((3.6,0)\) and a covertex at \((0, 7.7)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (7.31, 3.93):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  4. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-1.1,0)\) and \((1.1,0)\) and a covertex at \((0, 6)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (5.53, 2.53):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  5. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-4.2,0)\) and \((4.2,0)\) and a covertex at \((0, 4)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (3.17, 3.35):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  6. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+3)^2}{25}+\frac{(y+4)^2}{36}=1 & B&:~~\frac{(x+3)^2}{25}+\frac{(y-4)^2}{36}=1 & C&:~~\frac{(x-3)^2}{25}+\frac{(y+4)^2}{36}=1 & D&:~~\frac{(x-3)^2}{25}+\frac{(y-4)^2}{36}=1 \\ E&:~~\frac{(x+4)^2}{25}+\frac{(y+3)^2}{36}=1 & F&:~~\frac{(x+4)^2}{25}+\frac{(y-3)^2}{36}=1 & G&:~~\frac{(x-4)^2}{25}+\frac{(y+3)^2}{36}=1 & H&:~~\frac{(x-4)^2}{25}+\frac{(y-3)^2}{36}=1 \\ I&:~~\frac{(x+3)^2}{36}+\frac{(y+4)^2}{25}=1 & J&:~~\frac{(x+3)^2}{36}+\frac{(y-4)^2}{25}=1 & K&:~~\frac{(x-3)^2}{36}+\frac{(y+4)^2}{25}=1 & L&:~~\frac{(x-3)^2}{36}+\frac{(y-4)^2}{25}=1 \\ M&:~~\frac{(x+4)^2}{36}+\frac{(y+3)^2}{25}=1 & N&:~~\frac{(x+4)^2}{36}+\frac{(y-3)^2}{25}=1 & O&:~~\frac{(x-4)^2}{36}+\frac{(y+3)^2}{25}=1 & P&:~~\frac{(x-4)^2}{36}+\frac{(y-3)^2}{25}=1 \\ \end{align}\]

    Equation



    Solution


  7. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+2)^2}{36}+\frac{(y+3)^2}{64}=1 & B&:~~\frac{(x+2)^2}{36}+\frac{(y-3)^2}{64}=1 & C&:~~\frac{(x-2)^2}{36}+\frac{(y+3)^2}{64}=1 & D&:~~\frac{(x-2)^2}{36}+\frac{(y-3)^2}{64}=1 \\ E&:~~\frac{(x+3)^2}{36}+\frac{(y+2)^2}{64}=1 & F&:~~\frac{(x+3)^2}{36}+\frac{(y-2)^2}{64}=1 & G&:~~\frac{(x-3)^2}{36}+\frac{(y+2)^2}{64}=1 & H&:~~\frac{(x-3)^2}{36}+\frac{(y-2)^2}{64}=1 \\ I&:~~\frac{(x+2)^2}{64}+\frac{(y+3)^2}{36}=1 & J&:~~\frac{(x+2)^2}{64}+\frac{(y-3)^2}{36}=1 & K&:~~\frac{(x-2)^2}{64}+\frac{(y+3)^2}{36}=1 & L&:~~\frac{(x-2)^2}{64}+\frac{(y-3)^2}{36}=1 \\ M&:~~\frac{(x+3)^2}{64}+\frac{(y+2)^2}{36}=1 & N&:~~\frac{(x+3)^2}{64}+\frac{(y-2)^2}{36}=1 & O&:~~\frac{(x-3)^2}{64}+\frac{(y+2)^2}{36}=1 & P&:~~\frac{(x-3)^2}{64}+\frac{(y-2)^2}{36}=1 \\ \end{align}\]

    Equation



    Solution


  8. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+2)^2}{9}+\frac{(y+6)^2}{16}=1 & B&:~~\frac{(x+2)^2}{9}+\frac{(y-6)^2}{16}=1 & C&:~~\frac{(x-2)^2}{9}+\frac{(y+6)^2}{16}=1 & D&:~~\frac{(x-2)^2}{9}+\frac{(y-6)^2}{16}=1 \\ E&:~~\frac{(x+6)^2}{9}+\frac{(y+2)^2}{16}=1 & F&:~~\frac{(x+6)^2}{9}+\frac{(y-2)^2}{16}=1 & G&:~~\frac{(x-6)^2}{9}+\frac{(y+2)^2}{16}=1 & H&:~~\frac{(x-6)^2}{9}+\frac{(y-2)^2}{16}=1 \\ I&:~~\frac{(x+2)^2}{16}+\frac{(y+6)^2}{9}=1 & J&:~~\frac{(x+2)^2}{16}+\frac{(y-6)^2}{9}=1 & K&:~~\frac{(x-2)^2}{16}+\frac{(y+6)^2}{9}=1 & L&:~~\frac{(x-2)^2}{16}+\frac{(y-6)^2}{9}=1 \\ M&:~~\frac{(x+6)^2}{16}+\frac{(y+2)^2}{9}=1 & N&:~~\frac{(x+6)^2}{16}+\frac{(y-2)^2}{9}=1 & O&:~~\frac{(x-6)^2}{16}+\frac{(y+2)^2}{9}=1 & P&:~~\frac{(x-6)^2}{16}+\frac{(y-2)^2}{9}=1 \\ \end{align}\]

    Equation



    Solution


  9. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+2)^2}{9}+\frac{(y+4)^2}{25}=1 & B&:~~\frac{(x+2)^2}{9}+\frac{(y-4)^2}{25}=1 & C&:~~\frac{(x-2)^2}{9}+\frac{(y+4)^2}{25}=1 & D&:~~\frac{(x-2)^2}{9}+\frac{(y-4)^2}{25}=1 \\ E&:~~\frac{(x+4)^2}{9}+\frac{(y+2)^2}{25}=1 & F&:~~\frac{(x+4)^2}{9}+\frac{(y-2)^2}{25}=1 & G&:~~\frac{(x-4)^2}{9}+\frac{(y+2)^2}{25}=1 & H&:~~\frac{(x-4)^2}{9}+\frac{(y-2)^2}{25}=1 \\ I&:~~\frac{(x+2)^2}{25}+\frac{(y+4)^2}{9}=1 & J&:~~\frac{(x+2)^2}{25}+\frac{(y-4)^2}{9}=1 & K&:~~\frac{(x-2)^2}{25}+\frac{(y+4)^2}{9}=1 & L&:~~\frac{(x-2)^2}{25}+\frac{(y-4)^2}{9}=1 \\ M&:~~\frac{(x+4)^2}{25}+\frac{(y+2)^2}{9}=1 & N&:~~\frac{(x+4)^2}{25}+\frac{(y-2)^2}{9}=1 & O&:~~\frac{(x-4)^2}{25}+\frac{(y+2)^2}{9}=1 & P&:~~\frac{(x-4)^2}{25}+\frac{(y-2)^2}{9}=1 \\ \end{align}\]

    Equation



    Solution


  10. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+2)^2}{25}+\frac{(y+4)^2}{36}=1 & B&:~~\frac{(x+2)^2}{25}+\frac{(y-4)^2}{36}=1 & C&:~~\frac{(x-2)^2}{25}+\frac{(y+4)^2}{36}=1 & D&:~~\frac{(x-2)^2}{25}+\frac{(y-4)^2}{36}=1 \\ E&:~~\frac{(x+4)^2}{25}+\frac{(y+2)^2}{36}=1 & F&:~~\frac{(x+4)^2}{25}+\frac{(y-2)^2}{36}=1 & G&:~~\frac{(x-4)^2}{25}+\frac{(y+2)^2}{36}=1 & H&:~~\frac{(x-4)^2}{25}+\frac{(y-2)^2}{36}=1 \\ I&:~~\frac{(x+2)^2}{36}+\frac{(y+4)^2}{25}=1 & J&:~~\frac{(x+2)^2}{36}+\frac{(y-4)^2}{25}=1 & K&:~~\frac{(x-2)^2}{36}+\frac{(y+4)^2}{25}=1 & L&:~~\frac{(x-2)^2}{36}+\frac{(y-4)^2}{25}=1 \\ M&:~~\frac{(x+4)^2}{36}+\frac{(y+2)^2}{25}=1 & N&:~~\frac{(x+4)^2}{36}+\frac{(y-2)^2}{25}=1 & O&:~~\frac{(x-4)^2}{36}+\frac{(y+2)^2}{25}=1 & P&:~~\frac{(x-4)^2}{36}+\frac{(y-2)^2}{25}=1 \\ \end{align}\]

    Equation



    Solution


  11. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 6.6 meters apart, and she uses a string that is 13 meters long (ignoring the amount needed to tie around the stake).

    How long is the major axis of the generated ellipse?


    Solution


  12. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 9 meters apart, and she uses a string that is 10.6 meters long (ignoring the amount needed to tie around the stake).

    How long is the major axis of the generated ellipse?


    Solution


  13. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 14.4 meters apart, and she uses a string that is 19.4 meters long (ignoring the amount needed to tie around the stake).

    How long is the major axis of the generated ellipse?


    Solution


  14. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 12.8 meters apart, and she uses a string that is 16 meters long (ignoring the amount needed to tie around the stake).

    How long is the major axis of the generated ellipse?


    Solution


  15. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 7.2 meters apart, and she uses a string that is 12 meters long (ignoring the amount needed to tie around the stake).

    How long is the major axis of the generated ellipse?


    Solution


  16. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 2.6 meters apart, and she uses a string that is 17 meters long (ignoring the amount needed to tie around the stake).

    How long is the minor axis of the generated ellipse?


    Solution


  17. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 11.2 meters apart, and she uses a string that is 13 meters long (ignoring the amount needed to tie around the stake).

    How long is the minor axis of the generated ellipse?


    Solution


  18. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 7.2 meters apart, and she uses a string that is 17 meters long (ignoring the amount needed to tie around the stake).

    How long is the minor axis of the generated ellipse?


    Solution


  19. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 2.6 meters apart, and she uses a string that is 17 meters long (ignoring the amount needed to tie around the stake).

    How long is the minor axis of the generated ellipse?


    Solution


  20. Question

    A gardener is making an ellipse using the pins and string method. She places the two stakes (pins) 8 meters apart, and she uses a string that is 11.6 meters long (ignoring the amount needed to tie around the stake).

    How long is the minor axis of the generated ellipse?


    Solution


  21. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 10.6 meters and the length of the minor axis to be 5.6 meters.

    How long should the string be?


    Solution


  22. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 14.6 meters and the length of the minor axis to be 9.6 meters.

    How long should the string be?


    Solution


  23. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 17 meters and the length of the minor axis to be 15 meters.

    How long should the string be?


    Solution


  24. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 14.6 meters and the length of the minor axis to be 11 meters.

    How long should the string be?


    Solution


  25. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 19.4 meters and the length of the minor axis to be 13 meters.

    How long should the string be?


    Solution


  26. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 19.4 meters and the length of the minor axis to be 14.4 meters.

    How far apart should the stakes be placed?


    Solution


  27. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 19.4 meters and the length of the minor axis to be 13 meters.

    How far apart should the stakes be placed?


    Solution


  28. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 17.8 meters and the length of the minor axis to be 7.8 meters.

    How far apart should the stakes be placed?


    Solution


  29. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 19.4 meters and the length of the minor axis to be 14.4 meters.

    How far apart should the stakes be placed?


    Solution


  30. Question

    A gardener is making an ellipse using the pins and string method. She wants the length of the major axis to be 12.2 meters and the length of the minor axis to be 12 meters.

    How far apart should the stakes be placed?


    Solution


  31. Question

    The following equation (in polynomial form) represents an ellipse.

    \[9x^2-36x+16y^2-32y-92=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  32. Question

    The following equation (in polynomial form) represents an ellipse.

    \[16x^2-64x+36y^2+360y+388=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  33. Question

    The following equation (in polynomial form) represents an ellipse.

    \[4x^2-32x+36y^2-504y+1684=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  34. Question

    The following equation (in polynomial form) represents an ellipse.

    \[9x^2+18x+4y^2+56y+169=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  35. Question

    The following equation (in polynomial form) represents an ellipse.

    \[9x^2-18x+64y^2+640y+1033=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  36. Question

    It is easy to find the area of an ellipse. It is surprisingly difficult to find the perimeter (you need calculus).

    The area of an ellipse is pretty intuitive because an ellipse is a stretched circle. The ellipse has two different radii, so instead of \(A=\pi\cdot r\cdot r\) (for a circle), an ellipse’s area is found with the following equation.

    \[A=\pi a b\]

    where \(a\) is the length of the semi-major axis and \(b\) is the length of the semi-minor axis. Or, \(a\) and \(b\) are the largest and smallest radii of the ellipse.

    If an ellipse is drawn with a string of length \(10.6\) meters and two stakes placed \(9\) meters apart. What is the area of the ellipse in square meters?

    (The tolerance is \(\pm 0.01 ~\mathrm{m}^2\).)


    Solution


  37. Question

    It is easy to find the area of an ellipse. It is surprisingly difficult to find the perimeter (you need calculus).

    The area of an ellipse is pretty intuitive because an ellipse is a stretched circle. The ellipse has two different radii, so instead of \(A=\pi\cdot r\cdot r\) (for a circle), an ellipse’s area is found with the following equation.

    \[A=\pi a b\]

    where \(a\) is the length of the semi-major axis and \(b\) is the length of the semi-minor axis. Or, \(a\) and \(b\) are the largest and smallest radii of the ellipse.

    If an ellipse is drawn with a string of length \(17.8\) meters and two stakes placed \(7.8\) meters apart. What is the area of the ellipse in square meters?

    (The tolerance is \(\pm 0.01 ~\mathrm{m}^2\).)


    Solution


  38. Question

    It is easy to find the area of an ellipse. It is surprisingly difficult to find the perimeter (you need calculus).

    The area of an ellipse is pretty intuitive because an ellipse is a stretched circle. The ellipse has two different radii, so instead of \(A=\pi\cdot r\cdot r\) (for a circle), an ellipse’s area is found with the following equation.

    \[A=\pi a b\]

    where \(a\) is the length of the semi-major axis and \(b\) is the length of the semi-minor axis. Or, \(a\) and \(b\) are the largest and smallest radii of the ellipse.

    If an ellipse is drawn with a string of length \(12.2\) meters and two stakes placed \(2.2\) meters apart. What is the area of the ellipse in square meters?

    (The tolerance is \(\pm 0.01 ~\mathrm{m}^2\).)


    Solution


  39. Question

    It is easy to find the area of an ellipse. It is surprisingly difficult to find the perimeter (you need calculus).

    The area of an ellipse is pretty intuitive because an ellipse is a stretched circle. The ellipse has two different radii, so instead of \(A=\pi\cdot r\cdot r\) (for a circle), an ellipse’s area is found with the following equation.

    \[A=\pi a b\]

    where \(a\) is the length of the semi-major axis and \(b\) is the length of the semi-minor axis. Or, \(a\) and \(b\) are the largest and smallest radii of the ellipse.

    If an ellipse is drawn with a string of length \(19.4\) meters and two stakes placed \(14.4\) meters apart. What is the area of the ellipse in square meters?

    (The tolerance is \(\pm 0.01 ~\mathrm{m}^2\).)


    Solution


  40. Question

    It is easy to find the area of an ellipse. It is surprisingly difficult to find the perimeter (you need calculus).

    The area of an ellipse is pretty intuitive because an ellipse is a stretched circle. The ellipse has two different radii, so instead of \(A=\pi\cdot r\cdot r\) (for a circle), an ellipse’s area is found with the following equation.

    \[A=\pi a b\]

    where \(a\) is the length of the semi-major axis and \(b\) is the length of the semi-minor axis. Or, \(a\) and \(b\) are the largest and smallest radii of the ellipse.

    If an ellipse is drawn with a string of length \(17\) meters and two stakes placed \(7.2\) meters apart. What is the area of the ellipse in square meters?

    (The tolerance is \(\pm 0.01 ~\mathrm{m}^2\).)


    Solution


  41. Question

    It can be helpful to represent an ellipse in parametric form:

    \[\begin{align} x &= h+r_1 \cos(t)\\ y &= k+r_2 \sin(t) \end{align}\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following parametric systems, for \(0\le t<2\pi\), would give the graph above?

    \[\begin{align} A&:~~x=-3+2\cos(t)~~~~y=-4+5\sin(t) & B&:~~x=-3+2\cos(t)~~~~y=4+5\sin(t) \\\\ C&:~~x=3+2\cos(t)~~~~y=-4+5\sin(t) & D&:~~x=3+2\cos(t)~~~~y=4+5\sin(t) \\\\ E&:~~x=-4+2\cos(t)~~~~y=-3+5\sin(t) & F&:~~x=-4+2\cos(t)~~~~y=3+5\sin(t) \\\\ G&:~~x=4+2\cos(t)~~~~y=-3+5\sin(t) & H&:~~x=4+2\cos(t)~~~~y=3+5\sin(t) \\\\ I&:~~x=-3+5\cos(t)~~~~y=-4+2\sin(t) & J&:~~x=-3+5\cos(t)~~~~y=4+2\sin(t) \\\\ K&:~~x=3+5\cos(t)~~~~y=-4+2\sin(t) & L&:~~x=3+5\cos(t)~~~~y=4+2\sin(t) \\\\ M&:~~x=-4+5\cos(t)~~~~y=-3+2\sin(t) & N&:~~x=-4+5\cos(t)~~~~y=3+2\sin(t) \\\\ O&:~~x=4+5\cos(t)~~~~y=-3+2\sin(t) & P&:~~x=4+5\cos(t)~~~~y=3+2\sin(t) \end{align}\]

    Equation



    Solution


  42. Question

    It can be helpful to represent an ellipse in parametric form:

    \[\begin{align} x &= h+r_1 \cos(t)\\ y &= k+r_2 \sin(t) \end{align}\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following parametric systems, for \(0\le t<2\pi\), would give the graph above?

    \[\begin{align} A&:~~x=-2+5\cos(t)~~~~y=-4+6\sin(t) & B&:~~x=-2+5\cos(t)~~~~y=4+6\sin(t) \\\\ C&:~~x=2+5\cos(t)~~~~y=-4+6\sin(t) & D&:~~x=2+5\cos(t)~~~~y=4+6\sin(t) \\\\ E&:~~x=-4+5\cos(t)~~~~y=-2+6\sin(t) & F&:~~x=-4+5\cos(t)~~~~y=2+6\sin(t) \\\\ G&:~~x=4+5\cos(t)~~~~y=-2+6\sin(t) & H&:~~x=4+5\cos(t)~~~~y=2+6\sin(t) \\\\ I&:~~x=-2+6\cos(t)~~~~y=-4+5\sin(t) & J&:~~x=-2+6\cos(t)~~~~y=4+5\sin(t) \\\\ K&:~~x=2+6\cos(t)~~~~y=-4+5\sin(t) & L&:~~x=2+6\cos(t)~~~~y=4+5\sin(t) \\\\ M&:~~x=-4+6\cos(t)~~~~y=-2+5\sin(t) & N&:~~x=-4+6\cos(t)~~~~y=2+5\sin(t) \\\\ O&:~~x=4+6\cos(t)~~~~y=-2+5\sin(t) & P&:~~x=4+6\cos(t)~~~~y=2+5\sin(t) \end{align}\]

    Equation



    Solution


  43. Question

    It can be helpful to represent an ellipse in parametric form:

    \[\begin{align} x &= h+r_1 \cos(t)\\ y &= k+r_2 \sin(t) \end{align}\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following parametric systems, for \(0\le t<2\pi\), would give the graph above?

    \[\begin{align} A&:~~x=-2+3\cos(t)~~~~y=-4+8\sin(t) & B&:~~x=-2+3\cos(t)~~~~y=4+8\sin(t) \\\\ C&:~~x=2+3\cos(t)~~~~y=-4+8\sin(t) & D&:~~x=2+3\cos(t)~~~~y=4+8\sin(t) \\\\ E&:~~x=-4+3\cos(t)~~~~y=-2+8\sin(t) & F&:~~x=-4+3\cos(t)~~~~y=2+8\sin(t) \\\\ G&:~~x=4+3\cos(t)~~~~y=-2+8\sin(t) & H&:~~x=4+3\cos(t)~~~~y=2+8\sin(t) \\\\ I&:~~x=-2+8\cos(t)~~~~y=-4+3\sin(t) & J&:~~x=-2+8\cos(t)~~~~y=4+3\sin(t) \\\\ K&:~~x=2+8\cos(t)~~~~y=-4+3\sin(t) & L&:~~x=2+8\cos(t)~~~~y=4+3\sin(t) \\\\ M&:~~x=-4+8\cos(t)~~~~y=-2+3\sin(t) & N&:~~x=-4+8\cos(t)~~~~y=2+3\sin(t) \\\\ O&:~~x=4+8\cos(t)~~~~y=-2+3\sin(t) & P&:~~x=4+8\cos(t)~~~~y=2+3\sin(t) \end{align}\]

    Equation



    Solution


  44. Question

    It can be helpful to represent an ellipse in parametric form:

    \[\begin{align} x &= h+r_1 \cos(t)\\ y &= k+r_2 \sin(t) \end{align}\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following parametric systems, for \(0\le t<2\pi\), would give the graph above?

    \[\begin{align} A&:~~x=-5+2\cos(t)~~~~y=-7+3\sin(t) & B&:~~x=-5+2\cos(t)~~~~y=7+3\sin(t) \\\\ C&:~~x=5+2\cos(t)~~~~y=-7+3\sin(t) & D&:~~x=5+2\cos(t)~~~~y=7+3\sin(t) \\\\ E&:~~x=-7+2\cos(t)~~~~y=-5+3\sin(t) & F&:~~x=-7+2\cos(t)~~~~y=5+3\sin(t) \\\\ G&:~~x=7+2\cos(t)~~~~y=-5+3\sin(t) & H&:~~x=7+2\cos(t)~~~~y=5+3\sin(t) \\\\ I&:~~x=-5+3\cos(t)~~~~y=-7+2\sin(t) & J&:~~x=-5+3\cos(t)~~~~y=7+2\sin(t) \\\\ K&:~~x=5+3\cos(t)~~~~y=-7+2\sin(t) & L&:~~x=5+3\cos(t)~~~~y=7+2\sin(t) \\\\ M&:~~x=-7+3\cos(t)~~~~y=-5+2\sin(t) & N&:~~x=-7+3\cos(t)~~~~y=5+2\sin(t) \\\\ O&:~~x=7+3\cos(t)~~~~y=-5+2\sin(t) & P&:~~x=7+3\cos(t)~~~~y=5+2\sin(t) \end{align}\]

    Equation



    Solution


  45. Question

    It can be helpful to represent an ellipse in parametric form:

    \[\begin{align} x &= h+r_1 \cos(t)\\ y &= k+r_2 \sin(t) \end{align}\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following parametric systems, for \(0\le t<2\pi\), would give the graph above?

    \[\begin{align} A&:~~x=-2+5\cos(t)~~~~y=-3+6\sin(t) & B&:~~x=-2+5\cos(t)~~~~y=3+6\sin(t) \\\\ C&:~~x=2+5\cos(t)~~~~y=-3+6\sin(t) & D&:~~x=2+5\cos(t)~~~~y=3+6\sin(t) \\\\ E&:~~x=-3+5\cos(t)~~~~y=-2+6\sin(t) & F&:~~x=-3+5\cos(t)~~~~y=2+6\sin(t) \\\\ G&:~~x=3+5\cos(t)~~~~y=-2+6\sin(t) & H&:~~x=3+5\cos(t)~~~~y=2+6\sin(t) \\\\ I&:~~x=-2+6\cos(t)~~~~y=-3+5\sin(t) & J&:~~x=-2+6\cos(t)~~~~y=3+5\sin(t) \\\\ K&:~~x=2+6\cos(t)~~~~y=-3+5\sin(t) & L&:~~x=2+6\cos(t)~~~~y=3+5\sin(t) \\\\ M&:~~x=-3+6\cos(t)~~~~y=-2+5\sin(t) & N&:~~x=-3+6\cos(t)~~~~y=2+5\sin(t) \\\\ O&:~~x=3+6\cos(t)~~~~y=-2+5\sin(t) & P&:~~x=3+6\cos(t)~~~~y=2+5\sin(t) \end{align}\]

    Equation



    Solution


  46. Question

    Consider the ellipse represented by the parametric system below (with \(0\le t < 2\pi\)): \[\begin{align} x &= 5\cos(t) \\ y &= 7\sin(t) \end{align}\]

    The parameter \(t\) is called the eccentric anomaly. Let’s consider the point on the ellipse when \(t=0.88\).

    plot of chunk unnamed-chunk-2

    That point with Cartesian coordinates \((5\cos(0.88), 7\sin(0.88))\) also has polar coordinates \((r,\phi)\), where \(r\) is the distance from origin and \(\phi\) is the angle counterclockwise from the positive \(x\) axis.

    Find \(\phi\). The tolerance is \(\pm 0.01\) radians.


    Solution


  47. Question

    Consider the ellipse represented by the parametric system below (with \(0\le t < 2\pi\)): \[\begin{align} x &= 5\cos(t) \\ y &= 6\sin(t) \end{align}\]

    The parameter \(t\) is called the eccentric anomaly. Let’s consider the point on the ellipse when \(t=0.78\).

    plot of chunk unnamed-chunk-2

    That point with Cartesian coordinates \((5\cos(0.78), 6\sin(0.78))\) also has polar coordinates \((r,\phi)\), where \(r\) is the distance from origin and \(\phi\) is the angle counterclockwise from the positive \(x\) axis.

    Find \(\phi\). The tolerance is \(\pm 0.01\) radians.


    Solution


  48. Question

    Consider the ellipse represented by the parametric system below (with \(0\le t < 2\pi\)): \[\begin{align} x &= 4\cos(t) \\ y &= 5\sin(t) \end{align}\]

    The parameter \(t\) is called the eccentric anomaly. Let’s consider the point on the ellipse when \(t=0.47\).

    plot of chunk unnamed-chunk-2

    That point with Cartesian coordinates \((4\cos(0.47), 5\sin(0.47))\) also has polar coordinates \((r,\phi)\), where \(r\) is the distance from origin and \(\phi\) is the angle counterclockwise from the positive \(x\) axis.

    Find \(\phi\). The tolerance is \(\pm 0.01\) radians.


    Solution


  49. Question

    Consider the ellipse represented by the parametric system below (with \(0\le t < 2\pi\)): \[\begin{align} x &= 4\cos(t) \\ y &= 6\sin(t) \end{align}\]

    The parameter \(t\) is called the eccentric anomaly. Let’s consider the point on the ellipse when \(t=0.92\).

    plot of chunk unnamed-chunk-2

    That point with Cartesian coordinates \((4\cos(0.92), 6\sin(0.92))\) also has polar coordinates \((r,\phi)\), where \(r\) is the distance from origin and \(\phi\) is the angle counterclockwise from the positive \(x\) axis.

    Find \(\phi\). The tolerance is \(\pm 0.01\) radians.


    Solution


  50. Question

    Consider the ellipse represented by the parametric system below (with \(0\le t < 2\pi\)): \[\begin{align} x &= 7\cos(t) \\ y &= 4\sin(t) \end{align}\]

    The parameter \(t\) is called the eccentric anomaly. Let’s consider the point on the ellipse when \(t=0.44\).

    plot of chunk unnamed-chunk-2

    That point with Cartesian coordinates \((7\cos(0.44), 4\sin(0.44))\) also has polar coordinates \((r,\phi)\), where \(r\) is the distance from origin and \(\phi\) is the angle counterclockwise from the positive \(x\) axis.

    Find \(\phi\). The tolerance is \(\pm 0.01\) radians.


    Solution


  51. Question

    An ellipse is the set of points with a constant sum of distances from two foci.

    A hyperbola is the set of points with a constant difference of distances from two foci.

    Let’s explore the idea that a hyberloa is a collection of points with a constant difference of distances. Consider the hyperbola below, centered at the origin with foci at \((-4.8,0)\) and \((4.8,0)\) and with vertices at \((-3.7,0)\) and \((3.7,0)\). We will begin by finding the distances (and their difference) from the foci to a vertex.

    plot of chunk unnamed-chunk-2

    Now let’s pick another point on the hyperbola: \((7.4,5.3)\)

    plot of chunk unnamed-chunk-3



    Solution


  52. Question

    An ellipse is the set of points with a constant sum of distances from two foci.

    A hyperbola is the set of points with a constant difference of distances from two foci.

    Let’s explore the idea that a hyberloa is a collection of points with a constant difference of distances. Consider the hyperbola below, centered at the origin with foci at \((-4.2,0)\) and \((4.2,0)\) and with vertices at \((-2.8,0)\) and \((2.8,0)\). We will begin by finding the distances (and their difference) from the foci to a vertex.

    plot of chunk unnamed-chunk-2

    Now let’s pick another point on the hyperbola: \((7.6,7.9)\)

    plot of chunk unnamed-chunk-3



    Solution


  53. Question

    An ellipse is the set of points with a constant sum of distances from two foci.

    A hyperbola is the set of points with a constant difference of distances from two foci.

    Let’s explore the idea that a hyberloa is a collection of points with a constant difference of distances. Consider the hyperbola below, centered at the origin with foci at \((-3.3,0)\) and \((3.3,0)\) and with vertices at \((-2.2,0)\) and \((2.2,0)\). We will begin by finding the distances (and their difference) from the foci to a vertex.

    plot of chunk unnamed-chunk-2

    Now let’s pick another point on the hyperbola: \((4.2,4)\)

    plot of chunk unnamed-chunk-3



    Solution


  54. Question

    An ellipse is the set of points with a constant sum of distances from two foci.

    A hyperbola is the set of points with a constant difference of distances from two foci.

    Let’s explore the idea that a hyberloa is a collection of points with a constant difference of distances. Consider the hyperbola below, centered at the origin with foci at \((-5.7,0)\) and \((5.7,0)\) and with vertices at \((-3.9,0)\) and \((3.9,0)\). We will begin by finding the distances (and their difference) from the foci to a vertex.

    plot of chunk unnamed-chunk-2

    Now let’s pick another point on the hyperbola: \((7.8,7.2)\)

    plot of chunk unnamed-chunk-3



    Solution


  55. Question

    An ellipse is the set of points with a constant sum of distances from two foci.

    A hyperbola is the set of points with a constant difference of distances from two foci.

    Let’s explore the idea that a hyberloa is a collection of points with a constant difference of distances. Consider the hyperbola below, centered at the origin with foci at \((-4.8,0)\) and \((4.8,0)\) and with vertices at \((-3,0)\) and \((3,0)\). We will begin by finding the distances (and their difference) from the foci to a vertex.

    plot of chunk unnamed-chunk-2

    Now let’s pick another point on the hyperbola: \((7,7.9)\)

    plot of chunk unnamed-chunk-3



    Solution